Potential barrier mimicking frequent location measurements in quantum Zeno dynamics

We show that quantum Zeno dynamics can be mimicked by the isolated evolution of an unobserved system in an effective potential. Monitoring frequently whether a particle remains in a region of space leads to the same wave-packet dynamics as placing the region on top of a potential barrier and letting the particle evolve on its own, without external couplings. We focus on very frequent but not continuous observation so that the particle abandons the initial region with some finite probability. The height of the barrier relative to the surroundings for a high frequency $\nu$ of the observations being mimicked is found numerically to be $h\nu /2$, where $h$ is Planck's constant.

Figure 1

Permanence probability $P\left(n{T}^{\prime }\right)$ in $[-1,1]$ in successive measurements at times $n{T}^{\prime }$ for a particle in the initial state $\psi (x^{\prime },0)\propto \text{rect}\left(x^{\prime }\right)exp(-x^{\prime 2})exp\left(i{p}^{\prime }{x}^{\prime }\right)[\text{rect}\left(x^{\prime }\right)=1$ if $x^{\prime }\in [-1,1]$ and 0 otherwise] for different frequencies $1/T^{\prime }$ of the measurements. In (a) the particle is initially at rest ($p^{\prime }=0$), and in (b) it is moving ($p^{\prime }=2$). Upper curves seem like solid curves because the dots are very close.

Figure 2

The potential $V_{\mathrm{\Omega }}\left(\nu \right)$ approaching in $\mathrm{\Omega }$ the Zeno dynamics of a free particle subjected to frequent observation of permanence in $\mathrm{\Omega }$.

Figure 3

Dashed curves: Permanence probability $P\left(t^{\prime }\right)$ in a barrier in $[-1,1]$ as a function of time for a particle of initial state $\psi (x^{\prime },0)\propto \text{rect}\left(x^{\prime }\right)exp(-x^{\prime 2})exp\left(i{p}^{\prime }{x}^{\prime }\right)$ for different values $V_0^{\prime }=\pi /T^{\prime }$ of the barrier depth. In (a) $p^{\prime }=0$; in (b) $p^{\prime }=2$. Figure 1 is reproduced in gray for comparison.

Figure 4

Gray curves: Snapshots of the probability density in the Zeno dynamics with observation in $[-1,1]$ with (a) $T^{\prime }=0.1$ and (b) $T^{\prime }=0.001$ for a particle in the initial state $\psi (x^{\prime },0)\propto \text{rect}\left(x^{\prime }\right)exp(-x^{\prime 2})exp\left(i{p}^{\prime }{x}^{\prime }\right),p^{\prime }=2$. Dashed curves: the same as the gray curves, but in the unobserved evolution with potential barriers in $[-1,1]$ of depth $V_0^{\prime }=\pi /T^{\prime }$.

Figure 5

For initial wave functions $\psi (x^{\prime },0)\propto \text{rect}\left(x^{\prime }\right)exp(-x^{\prime 2}/s^2)exp\left(i{p}^{\prime }{x}^{\prime }\right)$ with $s=1,p^{\prime }=0$ (open squares), $s=2,p^{\prime }=0$ (open circles), $s=1,p^{\prime }=1$ (triangles), and $s=1,p^{\prime }=2$ (closed circles), numerically evaluated depth $V_0^{\prime }$ of the potential barrier in $[-1,1]$ that gives the same lingering in probability in $[-1,1]$ at $t^{\prime }=1$ as in the Zeno dynamics with measurements in $[-1,1]$ for increasing values $1/T^{\prime }$ of the frequency of the measurements. The dashed line is the dimensionless depth $V_0^{\prime }=\pi /T^{\prime }$ of the barrier in Eq. (6).

Figure 6

(a) For a particle in the initial state $\psi (x^{\prime },0)\propto exp(-x^{\prime 2})exp\left(i{p}^{\prime }{x}^{\prime }\right)$ in $(-\infty ,1]$ and $\psi (x^{\prime },0)=0$ otherwise, with $p^{\prime }=4$, lingering probability $P\left(n{T}^{\prime }\right)$ in $(-\infty ,1]$ at the $n\mathrm{th}$ measurement at $n{T}^{\prime }$ with period $T^{\prime }=0.001$ (gray curve) and lingering probability $P\left(t^{\prime }\right)$ in $(-\infty ,1]$ in barriers in $(-\infty ,1]$ of different depths $V_0^{\prime }=\pi /2T^{\prime }$ (blue curve), $V_0^{\prime }=\pi /T^{\prime }$ (black curve), and $V_0^{\prime }=2\pi /T^{\prime }$ (cyan curve). (b) Snapshots of the probability density for the same particle in the Zeno dynamics with $T^{\prime }=0.001$ (gray curves) and for the unobserved evolution with potential barriers in $(-\infty ,1]$ of depths $V_0^{\prime }=\pi /2T^{\prime }$ (blue curve), $V_0=\pi /T^{\prime }$ (black curve), and $V_0=2\pi /T^{\prime }$ (cyan curve).

Figure 7

A potential and the potential $V_{\mathrm{\Omega }}\left(\nu \right)$ approaching in $\mathrm{\Omega }$ the Zeno dynamics of a particle subjected to the potential $V$ and frequent observations of permanence in $\mathrm{\Omega }$.

Figure 8

For a particle in the initial state $\psi (x^{\prime },0)\propto \text{rect}\left(x^{\prime }\right)exp[-{(x^{\prime }/0.5)}^2]$ subjected to the potential $V^{\prime }=10x^{\prime }$, (a) probability of permanence in $[-1,1]$, (b) expected value of position, and (c) expected value of momentum in $[-1,1]$, when the particle is observed to be in $[-1,1]$ with period $T^{\prime }=0.001$ (gray curves) and when it is not observed but subjected to the potential $V_{\mathrm{\Omega }}^{\prime }=10x^{\prime }$ in $[-1,1],V_{\mathrm{\Omega }}^{\prime }=10-\pi /T^{\prime }$ in $(1,\infty )$, and $V_{\mathrm{\Omega }}^{\prime }=-10-\pi /T^{\prime }$ in $(-\infty ,-1)$ (dashed curves).